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<b>let </b><font color="Maroon" title="c1">a</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:12"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/>
<a NAME="E4:12"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="power.html#E3:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0 or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><b>suppose </b></a><a NAME="E1:12_1_1_1"/><i><font color="Green" title="E11">A4</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:12_1_1_1"/><b>then </b><i><font color="Green" title="E12">A5</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T60" title="XREAL_1:th.60">XREAL_1:60</a></i>;<br/><b>thus </b><font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="power.html#E1:12_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="power.html#D1" target="_self" title="POWER:def.1">Def1</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E4:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="power.html#E2:12_1_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="power.html#D1" target="_self" title="POWER:def.1">Def1</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_2"><b>suppose </b></a><a NAME="E1:12_1_1_2"/><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> = 
0
<b>by </b><i><a class="txt" href="power.html#E4:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="power.html#T6" target="_self" title="POWER:th.6">Th6</a></i>
<br/>.= 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E4:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="power.html#E1:12_1_1_2"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="power.html#T6" target="_self" title="POWER:th.6">Th6</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_3"><b>suppose </b></a><a NAME="E1:12_1_1_3"/><i><font color="Green" title="E11">A7</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:12_1_1_3"/><b>then </b>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/><b>hence </b> <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> = 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="prepower.html#K4" title="PREPOWER:func.4">-Root</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p4">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="power.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="power.html#D1" target="_self" title="POWER:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font>
<b>by </b><i><a class="txt" href="power.html#E4:12"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="power.html#E1:12_1_1_3"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="power.html#D1" target="_self" title="POWER:def.1">Def1</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:12"/>
<font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="power.html#K1" title="POWER:func.1">-root</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
 ;<br/>


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